Abstract: An integral on the set of natural numbers is defined. If is a subset of such that the characteristic function of is evaluated to some number by a regular nonnegative summation matrix , then is defined to be defines a finitely additive measure on . If is a sequence which can be written as a linear combination of characteristic functions , where each sequence is evaluated by , then is defined to be . Finally the definition of the integral is naturally extended to , the class of sequences which can be approximated by linear combinations of characteristic functions [2, pp. 85-88]. It is shown that if and are two nonnegative regular matrices such that the convergence field of includes that of , then includes provided is normal. Finally for a nonnegative regular matrix , the spaces spanned by sequences such that is bounded and exists are studied. It is shown that if is greater than one, then the sequences in give rise to a set of bounded linear functionals on which are weak star dense in the dual of .